May  2015, 20(3): 833-852. doi: 10.3934/dcdsb.2015.20.833

Stochastic dynamics in a fluid--plate interaction model with the only longitudinal deformations of the plate

1. 

Department of Mechanics and Mathematics, Karazin Kharkov National University, Kharkov, 61022, Ukraine

2. 

Institut für Mathematik, Institut für Stochastik, Ernst Abbe Platz 2, 07737, Jena, Germany

Received  September 2013 Revised  February 2014 Published  January 2015

We consider a stochastically perturbed coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a random pullback attractor of finite fractal dimension. Our argument is based on some modification of the method of quasi-stability estimates recently developed for deterministic systems.
Citation: Igor Chueshov, Björn Schmalfuß. Stochastic dynamics in a fluid--plate interaction model with the only longitudinal deformations of the plate. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 833-852. doi: 10.3934/dcdsb.2015.20.833
References:
[1]

L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (1998).  doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

G. Avalos, The strong stability and instability of a fluid-structure semigroup,, Appl. Math. Optim., 55 (2007), 163.  doi: 10.1007/s00245-006-0884-z.  Google Scholar

[3]

G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417.  doi: 10.3934/dcdss.2009.2.417.  Google Scholar

[4]

V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, in Fluids and waves, (2007), 55.  doi: 10.1090/conm/440/08476.  Google Scholar

[5]

H. Bauer, Probability Theory,, Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author, (1991).  doi: 10.1515/9783110814668.  Google Scholar

[6]

P. Boxler, Stochastisch Zentrumsmannigfaltigkeiten,, Thesis, (1988).   Google Scholar

[7]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, Lecture Notes in Mathematics, (1977).  doi: 10.1007/BFb0087685.  Google Scholar

[8]

A. Chambolle, B. Desjardins, M. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate,, J. Math. Fluid Mech., 7 (2005), 368.  doi: 10.1007/s00021-004-0121-y.  Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002).   Google Scholar

[10]

I. Chueshov, Monotone Random Systems Theory and Applications,, Lecture Notes in Mathematics, (1779).  doi: 10.1007/b83277.  Google Scholar

[11]

I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, Math. Methods Appl. Sci., 34 (2011), 1801.  doi: 10.1002/mma.1496.  Google Scholar

[12]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469.  doi: 10.1007/s10884-004-4289-x.  Google Scholar

[13]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008).  doi: 10.1090/memo/0912.  Google Scholar

[14]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Well-posedness and long-time dynamics, (2010).  doi: 10.1007/978-0-387-87712-9.  Google Scholar

[15]

I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations,, J. Dynam. Differential Equations, 13 (2001), 355.  doi: 10.1023/A:1016684108862.  Google Scholar

[16]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model,, Commun. Pure Appl. Anal., 12 (2013), 1635.  doi: 10.3934/cpaa.2013.12.1635.  Google Scholar

[17]

I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, J. Differential Equations, 254 (2013), 1833.  doi: 10.1016/j.jde.2012.11.006.  Google Scholar

[18]

I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in System Modeling and Optimization (25th IFIP TC7 Conference, (2013), 328.  doi: 10.1007/978-3-642-36062-6_33.  Google Scholar

[19]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,, Second edition, (2011).  doi: 10.1007/978-0-387-09620-9.  Google Scholar

[20]

G. P. Galdi, C. G. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$,, Math. Ann., 331 (2005), 41.  doi: 10.1007/s00208-004-0573-7.  Google Scholar

[21]

M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid,, J. Math. Fluid Mech., 10 (2008), 388.  doi: 10.1007/s00021-006-0236-4.  Google Scholar

[22]

M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model,, Appl. Anal., 88 (2009), 1053.  doi: 10.1080/00036810903114841.  Google Scholar

[23]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988).   Google Scholar

[24]

N. D. Kopachevskii and Y. S. Pashkova, Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane,, Russian J. Math. Phys., 5 (1997), 459.   Google Scholar

[25]

J. E. Lagnese, Modelling and stabilization of nonlinear plates,, in Estimation and Control of Distributed Parameter Systems (Vorau, (1990), 247.   Google Scholar

[26]

J. E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], (1988).   Google Scholar

[27]

J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set,, J. Math. Pures Appl. (9), 85 (2006), 269.  doi: 10.1016/j.matpur.2005.08.001.  Google Scholar

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[29]

T. J. Pedley, The Fluid Mechanics of Large Blood Vessels,, Cambridge University Press, (1980).  doi: 10.1017/CBO9780511896996.  Google Scholar

[30]

K. Petersen, Ergodic Theory,, Cambridge Studies in Advanced Mathematics, (1983).  doi: 10.1017/CBO9780511608728.  Google Scholar

[31]

B. Schmalfuß, The random attractor of the stochastic Lorenz system,, Z. Angew. Math. Phys., 48 (1997), 951.  doi: 10.1007/s000330050074.  Google Scholar

[32]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[33]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach,, [2013 reprint of the 2001 original] [MR1928881], (2001).   Google Scholar

[34]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Reprint of the 1984 edition, (1984).   Google Scholar

[35]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, Second edition, (1995).   Google Scholar

[36]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).   Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems,, Springer Monographs in Mathematics, (1998).  doi: 10.1007/978-3-662-12878-7.  Google Scholar

[2]

G. Avalos, The strong stability and instability of a fluid-structure semigroup,, Appl. Math. Optim., 55 (2007), 163.  doi: 10.1007/s00245-006-0884-z.  Google Scholar

[3]

G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 417.  doi: 10.3934/dcdss.2009.2.417.  Google Scholar

[4]

V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model,, in Fluids and waves, (2007), 55.  doi: 10.1090/conm/440/08476.  Google Scholar

[5]

H. Bauer, Probability Theory,, Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author, (1991).  doi: 10.1515/9783110814668.  Google Scholar

[6]

P. Boxler, Stochastisch Zentrumsmannigfaltigkeiten,, Thesis, (1988).   Google Scholar

[7]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions,, Lecture Notes in Mathematics, (1977).  doi: 10.1007/BFb0087685.  Google Scholar

[8]

A. Chambolle, B. Desjardins, M. J. Esteban and C. Grandmont, Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate,, J. Math. Fluid Mech., 7 (2005), 368.  doi: 10.1007/s00021-004-0121-y.  Google Scholar

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, American Mathematical Society Colloquium Publications, (2002).   Google Scholar

[10]

I. Chueshov, Monotone Random Systems Theory and Applications,, Lecture Notes in Mathematics, (1779).  doi: 10.1007/b83277.  Google Scholar

[11]

I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, Math. Methods Appl. Sci., 34 (2011), 1801.  doi: 10.1002/mma.1496.  Google Scholar

[12]

I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, J. Dynam. Differential Equations, 16 (2004), 469.  doi: 10.1007/s10884-004-4289-x.  Google Scholar

[13]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping,, Mem. Amer. Math. Soc., 195 (2008).  doi: 10.1090/memo/0912.  Google Scholar

[14]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Well-posedness and long-time dynamics, (2010).  doi: 10.1007/978-0-387-87712-9.  Google Scholar

[15]

I. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations,, J. Dynam. Differential Equations, 13 (2001), 355.  doi: 10.1023/A:1016684108862.  Google Scholar

[16]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model,, Commun. Pure Appl. Anal., 12 (2013), 1635.  doi: 10.3934/cpaa.2013.12.1635.  Google Scholar

[17]

I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, J. Differential Equations, 254 (2013), 1833.  doi: 10.1016/j.jde.2012.11.006.  Google Scholar

[18]

I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in System Modeling and Optimization (25th IFIP TC7 Conference, (2013), 328.  doi: 10.1007/978-3-642-36062-6_33.  Google Scholar

[19]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,, Second edition, (2011).  doi: 10.1007/978-0-387-09620-9.  Google Scholar

[20]

G. P. Galdi, C. G. Simader and H. Sohr, A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q,q}$,, Math. Ann., 331 (2005), 41.  doi: 10.1007/s00208-004-0573-7.  Google Scholar

[21]

M. Grobbelaar-Van Dalsen, On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid,, J. Math. Fluid Mech., 10 (2008), 388.  doi: 10.1007/s00021-006-0236-4.  Google Scholar

[22]

M. Grobbelaar-Van Dalsen, A new approach to the stabilization of a fluid-structure interaction model,, Appl. Anal., 88 (2009), 1053.  doi: 10.1080/00036810903114841.  Google Scholar

[23]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs, (1988).   Google Scholar

[24]

N. D. Kopachevskii and Y. S. Pashkova, Small oscillations of a viscous fluid in a vessel bounded by an elastic membrane,, Russian J. Math. Phys., 5 (1997), 459.   Google Scholar

[25]

J. E. Lagnese, Modelling and stabilization of nonlinear plates,, in Estimation and Control of Distributed Parameter Systems (Vorau, (1990), 247.   Google Scholar

[26]

J. E. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], (1988).   Google Scholar

[27]

J. A. Langa and J. C. Robinson, Fractal dimension of a random invariant set,, J. Math. Pures Appl. (9), 85 (2006), 269.  doi: 10.1016/j.matpur.2005.08.001.  Google Scholar

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[29]

T. J. Pedley, The Fluid Mechanics of Large Blood Vessels,, Cambridge University Press, (1980).  doi: 10.1017/CBO9780511896996.  Google Scholar

[30]

K. Petersen, Ergodic Theory,, Cambridge Studies in Advanced Mathematics, (1983).  doi: 10.1017/CBO9780511608728.  Google Scholar

[31]

B. Schmalfuß, The random attractor of the stochastic Lorenz system,, Z. Angew. Math. Phys., 48 (1997), 951.  doi: 10.1007/s000330050074.  Google Scholar

[32]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations,, Applied Mathematical Sciences, (2002).  doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[33]

H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach,, [2013 reprint of the 2001 original] [MR1928881], (2001).   Google Scholar

[34]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Reprint of the 1984 edition, (1984).   Google Scholar

[35]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, Second edition, (1995).   Google Scholar

[36]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).   Google Scholar

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